A late double from replacement AJ Lam finally allowed the Blues to shake off their much-changed opponents, who rested several first-choice players.
Hooker Kurt Eklund barrelled through a vacant defence for the opening try at Eden Park and a bloodied Adrian Choat, who took a stray boot in the face for his troubles, touched down to help make it 14-9 at the interval.
But Bryn Gatland's unerring boot reduced the deficit to two points early in the second period as the five-eighth went four from four off the tee.
Otere Black was off the mark with his conversion from Zarn Sullivan's maiden Super Rugby try and, when Gatland had no such problems after Bailyn Sullivan crossed, the scored were level with an hour played.
Black re-established his composure and the Blues lead with his only penalty of the match and, added the extras when Hoskins Sotutu powered over off the back off a five-metre scrum.
The Chiefs were unable to rally again, knowing their campaign will ultimately be defined a week from now, and Lam made hay with his late salvo.